Circular Layer Calculator

Name: Circular Layer - Geometry Calculator
Author: Jürgen Kummer

Calculations with a circular layer or circular zone in a semicircle. A circular layer is the difference between two circular segments, it is made of two parallel chords in a circle, which are on the same side of the circle center (or s through the center, if i=0).
Enter three of the four values of radius and h, i, j and choose the number of decimal places. Then click Calculate. Angles are calculated and displayed in degrees, here you can convert angle units.

Formulas:

r = h + i + j

s = 2 \cdot \sqrt{2 \cdot r \cdot (h + j) - {(h + j)}^{2}}

t = 2 \cdot \sqrt{2 \cdot r \cdot j - j^{2}}

Θ_{s} = 2 \cdot \arccos (1 - \frac{(h + j)}{r})

Θ_{t} = 2 \cdot \arccos (1 - \frac{j}{r})

l = \frac{r \cdot (Θ_{s} - Θ_{t})}{2}

p = s + t + 2 \cdot l

A = \frac{[r^{2} \cdot (Θ_{s} - Θ_{t}) - s \cdot i + t (r - j)]}{2}

Radius, height, distances, chords and perimeter have the same unit (e.g. meter), the area has this unit squared (e.g. square meter). For the angles Θ see circular segment.

This circular layer is created when a circular segment with the height j is cut off from a circular segment with the height h+j of the same circle in the same direction and the bases of the two segments are parallel. For the calculation of the shape in which the circular layer extends beyond the center of the circle, i.e., this center lies within the circular layer, see circular central segment. The three-dimensional equivalent of a circular layer is a spherical segment, which can also be called spherical layer.
The circular layer is axially symmetrical to the straight line that runs perpendicular to the base and through the center of this shape, i.e., exactly through the height j of the truncated spherical segment. A circular layer has no other symmetries.

The term layer is used here for geometry in a similar way to the colloquial term, as a connected area between two other areas. Therefore, the term stripe was chosen for the annulus stripe instead of layer because the layer is interrupted there, although stripes are also likely to be perceived as connected. The circular layer could therefore also have been called a circular stripe. Naming geometric objects that do not have an established name is sometimes not easy, because they either sound strange or, when using a common term, can evoke false associations.

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